How Lagrangian states evolve into random waves

Ingremeau, Maxime; Rivera, Alejandro

Journal de l’École polytechnique — Mathématiques, Tome 9 (2022), pp. 177-212.

Dans cet article, nous considérons une variété riemannienne connexe, compacte, de courbure sectionnelle négative, et une famille d’états lagrangiens semi-classiques $f_h\left(x\right)=a\left(x\right)e^{i\varphi \left(x\right)/h}$ sur $X$. Pour une grande famille de phases $\varphi$, nous montrons que $f_h$ que l’on fait évoluer par l’équation de Schrödinger pendant un temps long ressemble à un champ aléatoire gaussien. Ceci peut être vu comme un analogue de la conjecture des ondes aléatoires de Berry pour les états lagrangiens.

In this paper, we consider a compact connected manifold $(X,g)$ of negative curvature, and a family of semi-classical Lagrangian states $f_h\left(x\right)=a\left(x\right)e^{i\varphi \left(x\right)/h}$ on $X$. For a wide family of phases $\varphi$, we show that $f_h$, when evolved by the semi-classical Schrödinger equation during a long time, resembles a random Gaussian field. This can be seen as an analogue of Berry’s random waves conjecture for Lagrangian states.

Classification : 35P99, 81B50, 81Q20
Keywords: Quantum chaos, semi-classical analysis, Berry’s conjecture, random waves
Mot clés : Chaos quantique, analyse semi-classique, conjecture de Berry, ondes aléatoires

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